Wiles proof of Fermat’s Last Theorem borrowed from from algebraic geometry and number theory (Wikipedia). Presumably these are equivalent representations for some problems. I imagine this as the grown-up equivalent of switching from Cartesian to polar coordinates to solve a freshman physics problem.
I wonder when AI will start exploring math-space, searching for more of these unexpected alignments and concordances.
It feels like that might happen soon, but I have no idea how it would work.
Update: from the Wikipedia article on Wiles proof, a much better example of what I’m trying to describe:
Japanese mathematician Goro Shimura, drawing on ideas posed by Yutaka Taniyama, conjectured that a connection might exist between two different mathematical objects then being studied, known as elliptic curves and modular forms.
Taniyama and Shimura posed the question whether, unknown to mathematicians, the two kinds of object were actually identical mathematical objects, just seen in different ways.
What I’m describing is an automated search for “identical mathematical objects … seen in different ways”.